Heat oil in a large skillet over medium heat. Cook the sausage until well browned, 5 to 7 minutes.
Stir in the onion; cook and stir until lightly browned, 5 to 7 minutes more.
Remove the sausages, and cut them into 1 1/2-inch to 2-inch pieces.
Pour the beef stock, tomatoes, tomato puree, and hot sauce into the skillet with the onion and bring the mixture to a simmer, scraping up any brown bits from the bottom.
Place the carrots into a slow cooker, followed by the garbanzo beans and kidney beans.
Arrange the sausage pieces on top.
Pour the sauce from the skillet into the slow cooker.